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\begin{document}

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\section*{tensor product of Banach spaces}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

\hypertarget{tensor_products_of_banach_spaces}{}\section*{{Tensor products of Banach spaces}}\label{tensor_products_of_banach_spaces}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{cross_norms}{Cross norms}\dotfill \pageref*{cross_norms} \linebreak
\noindent\hyperlink{schmidt_decomposition}{Schmidt decomposition}\dotfill \pageref*{schmidt_decomposition} \linebreak
\noindent\hyperlink{foundational_issues}{Foundational issues}\dotfill \pageref*{foundational_issues} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

There are various [[norms]] that may be placed on the [[tensor product]] of the underlying [[vector spaces]] of two [[Banach spaces]]; the result is not usually [[complete space|complete]], but of course we may take its [[completion]]. One of these, the \emph{projective tensor product}, makes [[Ban]] (the category of Banach spaces and [[short linear maps]]) into a [[closed symmetric monoidal category]], but there are others that still put useful structures on $Ban$. If we start with [[Hilbert spaces]], then there is a choice of norm that will make the result into a Hilbert space; then [[Hilb]] also becomes a closed symmetric monoidal category.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Let $V$ and $W$ be [[Banach spaces]], and let $V \otimes W$ be their [[tensor product]] as [[vector spaces]]. To define a tensor product of $V$ and $W$ as Banach spaces, we will place a [[norm]] on $V \otimes W$, making a [[normed vector space]]; the only difference in the following definitions is which norm to use. Then we take the completion $V {\displaystyle\hat{\otimes}} W$, which is a Banach space.

\begin{defn}
\label{}\hypertarget{}{}
Every element of $V \otimes W$ may be written (in many different ways) as a formal [[linear combination]] of formal tensor products of elements of $V$ and $W$ (suppressing the symbol $\otimes$):

\begin{displaymath}
\sum_i \alpha_i v_i w_i .
\end{displaymath}
Let the \textbf{projective cross norm} ${\|x\|_\pi}$ of an element $x$ of $V \otimes W$ be

\begin{displaymath}
{\|x\|_\pi} \coloneqq \inf \{ \sum_i {|\alpha_i|} {\|v_i\|_V} {\|w_i\|_W} \;|\; x = \sum_i \alpha_i v_i w_i \} .
\end{displaymath}
Then the \textbf{projective tensor product} $V {\displaystyle\hat{\otimes}_\pi} W$ of $V$ and $W$ is the completion of $V \otimes W$ under the projective cross norm.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
If $\lambda$ and $\mu$ are [[linear functionals]] on $V$ and $W$ (respectively), then $\lambda \otimes \mu$ is a linear functional on $V \otimes W$. Let the \textbf{injective cross norm} ${\|x\|_\epsilon}$ of an element $x$ of $V \otimes W$ be

\begin{displaymath}
{\|x\|_\epsilon} \coloneqq \sup \{ {|(\lambda \otimes \mu)x|} \;|\; {\|\lambda\|_{V^*}}, {\|\mu\|_{W^*}} \leq 1 \} .
\end{displaymath}
Then the \textbf{injective tensor product} $V {\displaystyle\hat{\otimes}_\epsilon} W$ of $V$ and $W$ is the completion of $V \otimes W$ under the injective cross norm.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
If $V$ and $W$ are [[Hilbert spaces]], then their norms determine and are determined by their [[inner products]], so let us discuss inner products. The elements of $V \otimes W$ are generated by elements of the form $v w$, so set

\begin{displaymath}
\langle{v_1 w_1, v_2 w_2}\rangle \coloneqq \langle{v_1, v_2}\rangle \langle{w_1, w_2}\rangle
\end{displaymath}
and extend by linearity. We write the norm of an element $x$ of the [[inner product space]] $V \otimes W$ as ${\|x\|_\sigma}$. Then the \textbf{tensor product} $V {\displaystyle\hat{\otimes}_\sigma} W$ of the Hilbert spaces $V$ and $W$ is the completion of $V \otimes W$ under this norm (or inner product).

\end{defn}
\hypertarget{cross_norms}{}\subsection*{{Cross norms}}\label{cross_norms}

Besides the specific norms defined above, we can define axioms of a reasonable norm on $V \otimes W$.

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{cross norm} on $V$ and $W$ is \emph{any} [[norm]] $\chi$ on $V \otimes W$ such that:

\begin{itemize}%
\item ${\|v \otimes w\|_\chi} = {\|v\|_V} {\|w\|_W}$ for any elements $v$ and $w$ of $V$ and $W$ (respectively);
\item ${\|\lambda \otimes \mu\|_{\chi^*}} = {\|\lambda\|_{V^*}} {\|\mu\|_{W^*}}$ for any [[bounded linear functionals]] $\lambda$ and $\mu$ on $V$ and $W$ (respectively).

\end{itemize}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A \textbf{uniform cross norm} is an operation that takes two Banach spaces and returns a norm on their algebraic tensor product, [[natural transformation|naturally]] in the two spaces. Equivalently, it's a [[functor]] $\chi\colon Ban \times Ban \to NVect$ that makes the following diagram commute (or fills it with a [[natural isomorphism]]):

\begin{displaymath}
\array { Ban \times Ban   & \overset{\chi}\rightarrow     & NVect \\
            \downarrow       &                               & \downarrow \\
            Vect \times Vect & \underset{\otimes}\rightarrow & Vect
}
\end{displaymath}
\end{defn}
A uniform cross norm is obviously desirable from the [[nPOV]], but does it meet the analysts' needs for a cross norm? Yes:

\begin{prop}
\label{}\hypertarget{}{}
A uniform cross norm assigns a cross norm to any two Banach spaces.

\end{prop}
The specific cross norms from the previous section qualify as much as possible:

\begin{prop}
\label{}\hypertarget{}{}
The projective and injective cross norms are uniform cross norms (and hence are in fact cross norms). The norm on the algebraic tensor product of two Hilbert spaces is also a cross norm.

\end{prop}
As far as I can tell, the Hilbert-space cross norm $\sigma$ doesn't apply to arbitrary Banach spaces, so it doesn't define a uniform cross norm as defined above; however, it does define a functor on $Hilb \times Hilb$, so it's as uniform as could be expected.

Looking only at the general theory of cross norms, the projective and injective cross norms appear naturally:

\begin{prop}
\label{topbot}\hypertarget{topbot}{}
If $\chi$ is any uniform cross norm, $V$ and $W$ are any Banach spaces, and $x$ is any element of $V \otimes W$, then

\begin{displaymath}
{\|x\|_\epsilon} \leq {\|x\|_\chi} \leq {\|x\|_\pi} .
\end{displaymath}
\end{prop}
That is, we have a [[poset]] of uniform cross norms, and the projective and injective cross norms are (respectively) the [[top]] and [[bottom]] of this poset.

Although $\sigma$ is not a uniform cross norm, it relates to $\epsilon$ and $\pi$ in the same way:

\begin{prop}
\label{}\hypertarget{}{}
If $V$ and $W$ are Hilbert spaces and $x$ is an element of $V \otimes W$, then

\begin{displaymath}
{\|x\|_\epsilon} \leq {\|x\|_\sigma} \leq {\|x\|_\pi} .
\end{displaymath}
\end{prop}
Actually, this would all be simpler if Propostion \ref{topbot} applied to \emph{arbitrary} cross norms and not just uniform ones. Perhaps it does. Or perhaps $\sigma$ extends to a uniform cross norm on all of $Ban$; that would also make things simpler. I don't know.

Of course, any cross norm $\chi$ on $V$ and $W$ allows us to form the Banach space $V {\displaystyle\hat{\otimes}_\chi} W$, which may reasonably be called a \textbf{tensor product} of $V$ and $W$; that's why we care.

\hypertarget{schmidt_decomposition}{}\subsection*{{Schmidt decomposition}}\label{schmidt_decomposition}

The \textbf{Schmidt decomposition} is a way of expressing a pure state in the tensor product of two Hilbert spaces in terms of states of the two components:

\begin{theorem}
\label{}\hypertarget{}{}
Let $A$ and $B$ be finite-dimensional Hilbert spaces. Let $|\psi\rangle$ be a [[pure state]] of $A \otimes B$. Then there exist [[orthonormal family|orthonormal families]] $\{ |i_A \rangle \}_i$ in $A$ and $\{ |i_B \rangle \}_i$ in $B$, and non-negative real numbers $\lambda_i$, such that

\begin{displaymath}
|\psi\rangle = \sum_i \lambda_i |i_A \rangle \otimes |i_B\rangle
\end{displaymath}
and $\sum_i \lambda_i^2 = 1$.

\end{theorem}
The numbers $\lambda_i$ are called the \textbf{Schmidt co-efficients} of $|\psi\rangle$, and the families $\{ |i_A\rangle \}$ and $\{ |i_B\rangle \}$ the \textbf{Schmidt bases} for $A$ and $B$.

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{Schmidt number} of $|\psi\rangle$ is the number of non-zero Schmidt coefficients of $|\psi\rangle$.

\end{defn}
\hypertarget{foundational_issues}{}\subsection*{{Foundational issues}}\label{foundational_issues}

We need the [[Hahn–Banach theorem]] for $\epsilon$ to be a cross norm; but $\sigma$ and $\pi$ work regardless. Possibly some of the other propositions rely on some other form of the [[axiom of choice]]; I haven't seen their proofs.

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[spatial tensor product]]
\item [[inductive tensor product]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item M. Nielsen and I. Chuang. \emph{Quantum Computation and Quantum Information}. Cambridge University Press. 2000.

\end{itemize}
Many facts taken from Wikipedia:

\begin{itemize}%
\item \href{https://en.wikipedia.org/wiki/Topological_tensor_product}{Topological tensor product} (which also discusses tensor products of [[locally convex spaces]]),
\item \href{https://en.wikipedia.org/wiki/Tensor_product_of_Hilbert_spaces}{Tensor product of Hilbert spaces}.

\end{itemize}
[[!redirects tensor product of a Banach space]] [[!redirects tensor products of a Banach space]] [[!redirects tensor product of Banach spaces]] [[!redirects tensor products of Banach spaces]]

[[!redirects tensor product of a Hilbert space]] [[!redirects tensor products of a Hilbert space]] [[!redirects tensor product of Hilbert spaces]] [[!redirects tensor products of Hilbert spaces]]

[[!redirects projective tensor product]] [[!redirects projective tensor products]] [[!redirects projective tensor norm]] [[!redirects projective tensor norms]] [[!redirects projective cross norm]] [[!redirects projective cross norms]]

[[!redirects injective tensor product]] [[!redirects injective tensor products]] [[!redirects injective tensor norm]] [[!redirects injective tensor norms]] [[!redirects injective cross norm]] [[!redirects injective cross norms]]

[[!redirects cross norm]] [[!redirects cross norms]] [[!redirects reasonable cross norm]] [[!redirects reasonable cross norms]] [[!redirects uniform cross norm]] [[!redirects uniform cross norms]]

[[!redirects Schmidt decomposition]] [[!redirects Schmidt decompositions]] [[!redirects Schmidt coefficient]] [[!redirects Schmidt coefficients]] [[!redirects Schmidt co-efficient]] [[!redirects Schmidt co-efficients]] [[!redirects Schmidt basis]] [[!redirects Schmidt bases]] [[!redirects Schmidt basises]] [[!redirects Schmidt number]] [[!redirects Schmidt numbers]]



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